# TU Wien Rendering #15 - Rendering Equation Properties ## Метаданные - **Канал:** Two Minute Papers - **YouTube:** https://www.youtube.com/watch?v=sg0pAwOSNGw - **Дата:** 29.04.2015 - **Длительность:** 13:56 - **Просмотры:** 5,835 ## Описание Equipped with the knowledge of BRDFs used for the two most common materials, we get a step closer to solve the Holy Rendering Equation. According to its infinite dimensional and singular properties, it is immensely difficult to solve, but to our surprise, with a few more tricks up our sleeve, we can make it happen. About the course: This course aims to give an overview of basic and state-of-the-art methods of rendering. Offline methods such as ray and path tracing, photon mapping and many other algorithms are introduced and various refinement are explained. The basics of the involved physics, such as geometric optics, surface and media interaction with light and camera models are outlined. The apparatus of Monte Carlo methods is introduced which is heavily used in several algorithms and its refinement in the form of stratified sampling and the Metropolis-Hastings method is explained. At the end of the course students should be familiar with common techniques in rendering and find their way around the current state-of-the-art of the field. Furthermore the exercises should deepen the attendees' understanding of the basic principles of light transport and enable them to write a simple rendering program themselves. These videos are the recordings of the lectures of 2015 at the Teschnische Universität Wien by Károly Zsolnai and Thomas Auzinger Course website and slides → http://www.cg.tuwien.ac.at/courses/Rendering/ Subscribe → http://www.youtube.com/subscription_center?add_user=keeroyz Web → https://cg.tuwien.ac.at/~zsolnai/ Twitter → https://twitter.com/karoly_zsolnai ## Содержание ### [0:00](https://www.youtube.com/watch?v=sg0pAwOSNGw) and of course if we have some super long live paths that are combinations of these then obviously the ray tracer the recursive ray tracer cannot take this into account why is that that's the big question let's go back to the illumination equation and imagine that i'm hitting a diffuse surface what do i tried to emphasize this earlier but i will emphasize again that i take the perfect reflection direction it doesn't matter if it's diffused or specular i take the perfect reflection direction well if i do this i have no idea about the surroundings of the object i have no idea what is for instance above this diffuse plane if there is some red object i don't shoot a radar in order to get some indirect illumination so i will have no idea about the surroundings of this object now if i switch to global illumination however there is this integration and the integra part of the integration is the incoming light the incoming gradients and how i can integrate this over the hemisphere is basically sending samples out in every direction in this hemisphere now if i do this then i will know about the surroundings of the object if there's a red wall or a red object in nearby or the desert nearby then i will have samples of the incoming light and therefore it will appear in the color of the object this is fundamental this is the very important way to understand why ray tracers are missing these effects now let's talk about the real deal the real physically based brdf models how does a diffuse vrdf look like it looks like this so fr is the brdf omega prime are incoming and outgoing directions x is a point on this object and these are probabilities now this is weird because i'm used to formulate so if i talk about diffusion i've seen l dot n that's a formula with variables and this is a freaking number what do i do with this number it's 1 over pi 1 does this even make sense can someone help me out with this bubbles don't mind a bit so this 1 over pi means that if this is a scalar if this probability distribution is a scalar remember that this is the distribution of the possible outgoing directions so imagine this scenario up here where you have an incoming direction and if i have a completely diffuse material it means that it will diffuse the incoming light in every direction so all possible outgoing directions on the hemisphere have the very same probability and if they then this should be a number then the whole brdf because whatever directions i specify here i will get the same probability and i can scale this one over pi with rho which is the albedo of the material because not all materials reflect all light in fact most or if not all of the materials we know absorb some amount of light so this is again a number this can be wavelength dependent because it depends how much you absorb on the red channel how much absorb on the blue channel but this can be potentially zero and then you have a black body something that absorbs everything so you can call you can you can change the color of the object if i'm not using the right terms but i'd like to remain intuitive so the albedo is going to give you the color of the object and this we can specify that zero okay the next question is this a ### [4:07](https://www.youtube.com/watch?v=sg0pAwOSNGw&t=247s) Is this a Probability Distribution Function probability distribution function of course it is why because it's because it integrates to 1. there are some other rules that we're going to disregard with respect to probability distribution functions how much does it integrate to this integrates to 1. why what does the engineer guy say well 1 over pi integrated from 0 to pi what does it mean i have a rectangle that is that has the height of 1 over pi and it has the width of pi what is the area of the rectangle let's multiply these two sides so it's a times b a is pi b is one over pi just multiply by two and you'll get one so this is indeed a probability distribution function good to go ### [4:54](https://www.youtube.com/watch?v=sg0pAwOSNGw&t=294s) Specular what about specular brdfs these are what describe mirrors how can i write such a brdf it's a bit trickier because it is fundamentally different than just diffuse materials why they don't diffuse incoming lighting in all possible directions what is possible is only one outgoing direction i've seen only one thing in the mirror not the mixture of everything like on the walls so this means that one outgoing direction is going to have a probability of one and every single other choices have a zero probability so this is indeed a probabilistic model that can be described by a delta distribution means that one guy has a probability of one and everyone else has zero so it's like elections in a dictatorship is this a probability distribution function it is but i put an asterisk there because i'm going to talk a bit more about this but let's say for now that it is because this is one for one incoming direction and zero everywhere else so we have the one that we are looking for ### [6:14](https://www.youtube.com/watch?v=sg0pAwOSNGw&t=374s) Glossy Drds and there are also glossy vrdfs we haven't been really talking about this in the first lecture of mine there was some vrf which was called spread on one of these images but i asked you to forget this term immediately glossy is the mixture of the two so it is not like a mirror but it's not like a completely diffused material so there is some view dependence in diffuse materials they are completely view independent mirrors are completely view dependent so it's like a mixture of the two it is possible that there are some ### [6:50](https://www.youtube.com/watch?v=sg0pAwOSNGw&t=410s) Glossy Materials glossy materials in this scene can you find them raise your hand if you see at least one of you okay yes how about the cupboard excellent yes anything else just shout at me anymore do you mean this no the floor but the cooking field glass slow from the oh yeah exactly that's also glossy so there is many examples i think the question would be what is not glossy in this scene the better this would be the better question and the table you are sitting at is also glossy it is a bit view dependent but it's not a mirror but it's not completely diffused and it also transfers the caustics so it has some diffusivity okay next question is if it looks good but the mathematician guy asks how accurate is this now we have these two images one of these is generated by means of global illumination solving this equation and the other one is a photograph do you know which is which raise your hand if so okay one person two okay i'm gonna spoil all the fun entire solution okay so look at this part so this is the difference that you can see for instance because this is an actual box that the guys put together at the corner university and you cannot only see the box in the photograph but what is next to the box whereas in global illumination these surroundings are not modeled just the kernel box itself so we can conclude that yes this can be distinguished from a photograph but if you look at the actual scene it is very difficult and if everything is perfectly implemented then this is so close to physical reality that it is literally indistinguishable so this is really amazing that we can do this whatever you see out there in the world we can model with this equation there are exceptions because there are wave effects such as diffraction and stuff like that but these are very rare i mean there are butterflies who look the way they look because of interference and then these effects but 99 of what you see can be modeled with this equation and the rest can be handled by more sophisticated methods so back to this previous question what ### [10:09](https://www.youtube.com/watch?v=sg0pAwOSNGw&t=609s) What Is the Dimensionality of the Rendering Equation is the dimensionality of the rendering equation let's try to think it through and we will see so let's for just for now imagine that i shoot the ray out from the camera and i hit the diffuse object i need to sample this hemisphere exhaustively this is not how i will evaluate the algorithm but technically this is what i need to do all possible outgoing directions have the same probability so i need to shoot these outgoing rays many of them now i will hit more diffuse objects after the first bounce and i have to exhaustively sample all of these as well and if i take this other a i also have to do this and so on and so on until how many bounces we have concluded previously that we have to take into consideration an infinite number of boxes so this is definitely very difficult because the incoming light that i am sampling the hemisphere for is another rendering equation so imagine that this li you can insert another one of these equations but that equation will also contain this integral and this li and that's another rendering equation so it's an infinitely large sequence of integrals therefore this is infinite dimensional now i told you before that this is also singular this is not such a bad thing but this is because of the possibility of specular brdfs the specular brdf is some kind of a delta distribution and delta distributions are not really functions so in signals processing you may have studied this function and the first thing that they tell you about is that this is not a function this can be defined in terms of a limit so you can for instance imagine like a gaussian curve and you start pushing this gaussian curve from two sides therefore this is going to be a larger and thinner and thinner spike and you do this until you have an infinitely thin spike now if you check it for the properties of a function you will get something that has nothing to do with the function that's a singularity there is an infinitely quick jump from zero to one in there and we need to handle this somehow because we can take into consideration functions we can integrate functions so let's just solve this trivially by handling this specular inter-reflection explicitly what does it mean this means that if you have an incoming direction you're not going to play with probabilities you are just going to grab like in a ray tracer the perfect reflection direction as an outgoing direction no probabilities nothing a beauty break we have some scenario which is ray tracing because of different things because the image you create by means of ray facing but there's literally one ray of light being reflected here many times so awesome laser experiments with lux render we will try things out like this a bit later during the course and another example so it's amazing what we can do with these algorithms --- *Источник: https://ekstraktznaniy.ru/video/14997*